# Small birhombatodecachoron

Small birhombatodecachoron | |
---|---|

Rank | 4 |

Type | Isogonal |

Notation | |

Bowers style acronym | Sabred |

Coxeter diagram | ao3ob3bo3oa&#zc (1/2 < b:a < 3) |

Elements | |

Cells | 30 tetragonal disphenoids, 10 octahedra, 20 triangular antiprisms, 40 triangular antipodiums |

Faces | 20+40+40 triangles, 120+120 isosceles triangles |

Edges | 60+120+120 |

Vertices | 60 |

Vertex figure | Apex-truncated augmented triangular prism |

Measures (based on two small rhombated pentachora of edge length 1) | |

Edge lengths | Lacing edges (120): |

Base edge 1 (60): 1 | |

Base edge 2 (120): 1 | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Sabred |

Regiment | Sabred |

Dual | Small biorthodecachoron |

Abstract & topological properties | |

Flag count | 4080 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}×2, order 240 |

Convex | Yes |

Nature | Tame |

The **small birhombatodecachoron** or **sabred** is a convex isogonal polychoron that consists of 10 octahedra, 20 triangular antiprisms, 40 triangular antipodiums, and 30 tetragonal disphenoids. 1 octahedron, 2 triangular antiprisms, 4 triangular antipodiums, and 2 tetragonal disphenoids join at each vertex.

It is one of a total of five distinct polychora (including two transitional cases) that can be obtained as the convex hull of two opposite small rhombated pentachora. In this case, the ratio between the edges of the small rhombated pentachoron a3o3b3o is between b:a = 1/2 (producing the rectified small prismatodecachoron) and b:a = 3 (producing the rectified decachoron). This includes the convex hull of two uniform small rhombated pentachora. The lacing edges generally have length .

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.29099.

## External links[edit | edit source]

- Bowers, Jonathan. "Pennic and Decaic Isogonals".

- Klitzing, Richard. "sabred".
- Wikipedia contributors. "Cantellated 5-cell#Related polytopes".